Measurements of ultrasonic phase velocities and attenuation
of slow waves in cellular aluminum foams as cancellous
bone-mimicking phantoms
Chan Zhang (
) and Lawrence H. Le (
)a)
Department of Radiology and Diagnostic Imaging, University of Alberta, Edmonton, Alberta T6G 2B7,
Canada
Rui Zheng (
)
Department of Physics, University of Alberta, Edmonton, Alberta T6G 2G7, Canada
Dean Ta (
)
Department of Electronic Engineering, Fudan University, Shanghai, China
Edmond Lou (
)
Department of Biomedical Engineering, University of Alberta, Edmonton, Alberta T6G 2V2, Canada
(Received 6 January 2010; revised 14 February 2011; accepted 15 February 2011)
The water-saturated aluminum foams with an open network of interconnected ligaments were
investigated by ultrasonic transmission technique for the suitability as cancellous bone-mimicking
phantoms. The phase velocities and attenuation of nine samples covering three pores per inch (5,
10, and 20 PPI) and three aluminum volume fractions (5, 8, and 12% AVF) were measured over
a frequency range of 0.7–1.3 MHz. The ligament thickness and pore sizes of the phantoms and
low-density human cancellous bones are similar. A strong slow wave and a weak fast wave are
observed for all samples while the latter is not visible without significant amplification (100).
This study reports the characteristics of slow wave, whose speeds are less than the sound speed of
the saturating water and decrease mildly with AVF and PPI with an average 1469 m=s. Seven out
of nine samples show positive dispersion and the rest show minor negative dispersion. Attenuation increases with AVF, PPI, and frequency except for the 20 PPI samples, which exhibit nonincreasing attenuation level with fluctuations due to scattering. The phase velocities agree with
Biot’s porous medium theory. The RMSE is 16.0 m=s (1%) at n ¼ 1.5. Below and above this
value, the RMSE decreases mildly and rises sharply, respectively.
C 2011 Acoustical Society of America. [DOI: 10.1121/1.3562560]
V
I. INTRODUCTION
Osteoporosis is one of the most common skeletal disorders characterized by reduced bone mass, microstructural
deterioration, cortical bone thinning, and cancellous bone
perforation leading to fracture risks (Werner, 2005). The
prevalence of osteoporotic fractures of the wrists, hips, and
spines is a serious public health problem affecting about 200
million people worldwide especially in the aging population
(Lin and Lane, 2004). Currently, bone mass is measured by
bone mineral density (BMD) using dual energy x-ray absorptiometry (DXA) for bone quality assessment. However, clinical studies indicated that BMD values measured in normal
and osteoporotic bones showed considerable overlap (Westmacott, 1995; Legrand et al., 2000; Silverman, 2006); therefore, BMD measurements do not provide sufficient
information for accurate diagnosis. Osteoporosis still
remains one of the most prevalent undiagnosed diseases in
the world today (Nguyen et al., 2004). This is due to the failure to consider bone elasticity and trabecular microstructures
a)
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
J. Acoust. Soc. Am. 129 (5), May 2011
Pages: 3317–3326
by radiation-based methods (Langton et al., 1984; Rossman
et al., 1989; Ulrich et al., 1999). These mechanical and
structural properties could be potentially assessed by the
quantitative ultrasound (QUS) and Langton and Njeh (2004)
provided a good review of the method.
Cancellous bone is made up of interconnected network
of trabeculae with viscous marrow filling the pores. Because
osteoporosis is mostly manifested in the loss of cancellous
bone mass and the subsequent disruption of trabecular
microstructures (Langton et al., 1984), measurements of ultrasonic attenuation and velocity in cancellous bones have
been used to evaluate bone quality (Droin et al., 1998).
However, due to high porosity and heterogeneity of human
cancellous bones (Gibson, 1985), the underlying mechanisms of interactions between ultrasound and cancellous
bones are still not fully exploited or understood (Bossy
et al., 2007; Haı̈at et al., 2008a; Pakula et al., 2008; Wear,
2008a, 2009). Cancellous bone-mimicking phantoms are
controllably designed with some desirable characteristic
features similar to those of cancellous bones and allow independent study of the influence of these characteristics upon
the measurables such as velocity and attenuation. Clarke
et al. (1994) and Strelitzki et al. (1997) developed phantoms
0001-4966/2011/129(5)/3317/10/$30.00
C 2011 Acoustical Society of America
V
3317
Author's complimentary copy
PACS number(s): 43.80.Jz, 43.80.Qf, 43.20.Hq [CCC]
FIG. 1. (Color online) An aluminum
foam sample: (a) photo image, (b)
micrograph of a 20 PPI-12% AVF
foam sample, and (c) a schematic
diagram of the pore elongation (The
diagram was provided by ERG
Materials and Aerospace Corporation, Oakland, CA). The arrow indicates the direction of the ultrasound
beam in the experiment.
using liquid epoxy and randomly distributed granules (the
Leeds bone phantoms) to study the influence of porosity and
pore size on velocity dispersion and attenuation. Lee and
Choi (2009) studied the ultrasonic parameters with
polyacetal cuboid bone-mimicking phantoms with cylindrical pores running along the three orthogonal axes. Wear
(2005, 2008a) measured velocity dispersion and attenuation
in bone-mimicking phantoms consisting of two-dimensional
arrays of parallel nylon wires to simulate trabeculae. Wear
(2008b) also studied frequency-dependent backscatter from
quasi-parallel-nylon-wire phantoms. In place of parallel nylon wires, Wear (2009) repeated his study using cancellous
bone-mimicking phantoms with suspensions of randomly
oriented nylon wires in a marrow-mimicking medium.
In this work, commercial open-celled aluminum foams
are used as candidates of cancellous bone-mimicking phantoms where the aluminum ligaments simulate the trabeculae.
Ultrasonic reports with aluminum foams are so far not many.
Ji et al. (1998) did some pioneering studies of the water-saturated aluminum foams and identified both fast and slow
waves in transmission-through experiments when the ultrasound beam direction was along the pore elongation; Weaver
(1998) studied the modal density of the aluminum foams;
Lobkis and Weaver (2001) also studied the modal density,
diffusivity, and absorption of the aluminum foams; Lu et al.
(1999) studied sound absorption capacity of Alporas aluminum foam. Recently, Le et al. (2010) measured the tortuosity
of the aluminum foams using airborne ultrasound as a precursor to investigate the ultrasonic properties of the foams
further. The objective of this study is to investigate the frequency-dependent phase velocity, dispersion, and attenuation in water-saturated aluminum foams. The ultrasonic
properties of the foams will be discussed and their similarity
to those of cancellous bone will be explored. The measured
wave speeds will also be compared with the predictions by
Biot’s porous medium theory (Biot, 1956a,b).
II. MATERIALS AND METHODS
A. Cellular aluminum foams
The DUOCEL aluminum foams (ERG Materials and
Aerospace Corporation, Oakland, CA) are made of 6101 aluminum. The foams are solution heat treated and then artificially aged to a stable condition. The mechanical and
structural stability are both substantially improved by the
heat treatment. The foam has an open-celled structure with
an interconnected network of aluminum ligaments. The
foam properties are described by two parameters: the number of cells or pores per inch (PPI) and the aluminum volume
fraction (AVF). The matrix of pores and ligaments is completely repeatable. Figures 1(a) and 1(b) show the image and
micrograph of a 20 PPI-12% AVF foam sample, respectively. Figure 1(c) shows a schematic diagram of a preferred
elongation of the pore structures during the forming process
of the aluminum foam samples due to the primary expansion
directly from the balance between gravitational and pneumatic forces (Le et al., 2010). The anisotropy of the structural behavior in the elongated direction is around 5%–15%
greater than the other two orthogonal directions. We considered three PPIs: 5, 10, and 20, and each PPI has three available AVFs: 5, 8, and 12%. Tables I and II summarize the
structural and physical properties of the aluminum foams.
All the samples were pre-cut in 50.8 50.8 12.7 mm3 rectangular prisms. Prior to the experiments, the samples were
immersed in distilled water over 24 h to ensure the pores
were fully saturated. A few drops of detergent were added
Sample #
1
2
3
4
5
6
7
8
9
3318
PPI
AVF ( %)
cell size (mm)
Al.Th (mm)
Al.Sp (mm)
Tortuosity
5
5
8
12
5
8
12
5
8
12
5.08
0.26
0.41
0.61
0.13
0.21
0.31
0.07
0.11
0.16
4.82
4.67
4.47
2.41
2.33
2.23
1.20
1.16
1.11
1.003 6 0.001
1.015 6 0.007
1.022 6 0.002
1.026 6 0.002
1.026 6 0.001
1.029 6 0.002
1.030 6 0.002
1.032 6 0.002
1.032 6 0.003
10
20
J. Acoust. Soc. Am., Vol. 129, No. 5, May 2011
2.54
1.27
Zhang et al.: Dispersion and attenuation in bone phantoms
Author's complimentary copy
TABLE I. Structural properties of aluminum foams. PPI stands for pores per inch. AVF is the aluminum volume fraction, the ratio of aluminum volume to
foam volume. Porosity is given by 1-AVF. The mean cell size (mm) is equal to 25.4=PPI. Al.Th is the aluminum ligament thickness and calculated by mean
cell size AVF. Al.Sp is the ligament separation and is determined by mean cell size (1-AVF). Le et al. (2010) provides the tortuosity values.
TABLE II. Physical parameters of aluminum and water.
Young’s modulus of aluminum,a Es
Poisson’s ratio of aluminum,a ts
Density of aluminum,a qs
Bulk modulus of water,b Kf
Density of water,b qf
Viscosity of water @ 20 C,b g
a
68.9 GPa
0.33
2700 kg=m3
2.19 GPa
1000 kg=m3
0.0001 N s=m2
MatWeb, Online Materials Information Resource.
Ji et al. (1998).
b
into the water to decrease the surface tension; the samples
were occasionally shaken slightly underwater to eliminate
air bubbles in the pores.
B. Experimental setup
Figure 2 shows the ultrasonic immersion experimental
setup using pulse transmission technique. The experiment
used a pair of 12.7-mm-diameter 1.0 MHz unfocused broadband transducers (Panametrics V303-SU, Waltham, MA).
The transducers were mounted coaxially on a sample holder
with the foam samples placed between the two transducers
at near field distance from the transmitter. The ultrasound
beam was perpendicular to the pore elongation of the sample
as shown in Fig. 1(c) and the whole setup was immersed in
distilled water. The speed of sound in distilled water is temperature-dependent (Wear, 2009)
Vwater ¼ 1402:9 þ 4:835T 0:047016T 2
þ 0:00012725T 3
(1)
FIG. 2. (Color online) A schematic diagram of the immersion experimental
setup. The distance between transducers was about twice the focal length,
d 53 mm.
where T is the temperature in Celsius. Since the temperature
of water was approximately 19.5 C, Vwater was 1480 m=s.
The near field distance N is given by N ¼ D2=4k, where D is
the diameter of the transducer and k is the wavelength of
ultrasound in water (Cobbold, 2008). In our case, N is 26.4
mm and the distance between the transducers was set at
about twice the focal length (53 mm). Nine regions of interests (ROIs) were scanned for each sample and each ROI was
shifted 2 mm from the previous ROI and parallel to the
transducer’s surface while maintaining the ultrasound beam
well within the sample boundaries. An Olympus 5800 pulsereceiver (Panametrics, Waltham, MA) excited the transmitter; the transmitted signals were received and stored by a
LeCroy WaveSufer 422 digital oscilloscope (LeCroy, Chestnut Ridge, NY). The signals were sampled at 2 Gigasamples
per second at 8 bit resolution. Each of the digitized
J. Acoust. Soc. Am., Vol. 129, No. 5, May 2011
Zhang et al.: Dispersion and attenuation in bone phantoms
3319
Author's complimentary copy
FIG. 3. The ultrasound signals: (a) a
water pulse; (b) a sample signal of a
5 PPI-5% AVF foam sample (sample #1) where the box outlines the
region where a fast wave exists; (c)
the amplitude spectrum of the water
pulse; and (d) the amplitude spectrum of the sample signal.
(Le, 1998) to calculate the delay time. When absorption
exists, dispersion analysis is useful to examine the traveling
speeds of the frequency components via phase velocities
(Sachse and Pao, 1977)
c ðx Þ ¼
FIG. 4. The region within the box in Fig. 3(b) was amplified 100 times to
show the existence of a fast wave.
waveform was continuously averaged over 256 times to
increase signal-to-noise-ratio (SNR). The signals were later
decimated and a small time window of 301 points containing
the relevant waveforms was selected for further spectral
analysis with zero-padding.
C. Computation of group velocity, phase velocity, and
attenuation
For transmission technique, two signals are recorded
without and with the sample in the beam path. The former is
a reference signal through water (water pulse) and the latter
is a sample pulse. The speed of sound of the medium can be
determined by the group velocity via the substitution method
(Rossman et al., 1989)
V¼
h
h
Dt þ Vwater
(2)
where Dt is the delay time between the traveling times of the
reference points of the signals, and h is the sample thickness.
In this work, we used the arrival times of the envelope peaks
xh
DuðxÞ þ Vxh
water
(3)
where x is the angular frequency and Du(x) is the
unwrapped phase difference between the reference and sample spectra. However, a least-squares fitted Dulsf(x) is used
in Eq. (3) instead and is estimated by the linear least-squares
regression fit to the Du(x) within the frequency range of interest (Verhoef et al., 1985; Mujagic et al., 2008). Dispersion
is characterized by the slope of the linear least-squares
regression line fitted to the phase velocity versus frequency
data within the same frequency range.
Ultrasonic energy loss due to scattering, absorption, and
transmission is quantified by the attenuation coefficient, a
(in dB=cm),
aðxÞ ¼
20
log10 Aref ðxÞ Asample ðxÞ
d
(4)
where Aref (x) and Asample (x) are the spectra of the reference and sample signals. The attenuation was not corrected
for the transmission loss caused by the water-foam interfaces. When the attenuation is normalized by the relevant frequency band, we obtain the normalized broadband
ultrasound attenuation (nBUA) in dB=MHz=cm.
Since each sample was measured nine times, the average group velocities, phase velocities, and attenuation were
calculated and reported with standard deviations.
III. RESULTS
The water pulse and an example of the 5 PPI-5% AVF
(sample #1) sample pulse are shown in Figs. 3(a) and 3(b)
and their corresponding amplitude spectra in Figs. 3(c) and
3(d), respectively. The sample pulse was attenuated by 6 dB
compared to the water pulse. There was no observable distortion on the wave shape and dispersion, if there is any, was
extremely small; the scattered ultrasound between pores
within the foam sample is evidenced as small variations in
the tail of the main pulse. The portion of the signal
Sample #
1
2
3
4
5
6
7
8
9
Group Velocity (m=s)
Dispersion (m=s=MHz)
nBUA (dB=MHz=cm)
1473 6 2.8
1472 6 2.4
1466 6 4.2
1472 6 2.4
1463 6 2.4
1467 6 4.4
1472 6 2.4
1472 6 2.4
1468 6 4.2
5.35 6 5.56
12.69 6 3.07
7.69 6 4.80
1.52 6 1.62
1.12 6 0.95
5.83 6 1.47
2.98 6 1.44
5.34 6 2.18
15.01 6 2.81
4.2 6 0.7
5.5 6 1.0
6.5 6 0.7
3.5 6 0.6
3.8 6 1.2
4.1 6 1.1
1.5 6 0.9
1.4 6 0.6a
0.8 6 1.4
a
For the fixed 0.7–1.3 MHz frequency band, the nBUA value is a small negative number (0.8 6 0.5), which renders amplification instead of attenuation. We
recalculated the nBUA value using a 0.8–1.4 MHz frequency window.
3320
J. Acoust. Soc. Am., Vol. 129, No. 5, May 2011
Zhang et al.: Dispersion and attenuation in bone phantoms
Author's complimentary copy
TABLE III. Group velocities, dispersion, and nBUA of aluminum foams.
J. Acoust. Soc. Am., Vol. 129, No. 5, May 2011
FIG. 5. The phase velocity versus frequency curves: (a) 5 PPI, (b) 10 PPI,
and (c) 20 PPI samples.
increasing AVF, while the attenuation increases with
increasing AVF. The behavior becomes less predictable for
various PPI at fixed AVF. First, for the 5% AVF samples
(#1, #4, and #7), both phase velocities and attenuation
showed minor changes with different PPI (less than 1 m=s
Zhang et al.: Dispersion and attenuation in bone phantoms
3321
Author's complimentary copy
delineated by the box in Fig. 3(b) was amplified 100 times in
Fig. 4 to reveal a weak fast wave preceding the strong slow
wave (Biot, 1956a,b); Furthermore, the fast wave has a time
period of approximately 1.7 ls compared with 1 ls of the
slow wave, indicating the dominant frequency of fast wave
is about 40% lower than the dominant frequency of the slow
wave. The same observations apply to the other eight samples. Since the slow waves dominate the wave propagation,
the ultrasound properties of the aluminum foams are governed by the slow waves within the frequency range from
0.7 to 1.3 MHz. The group velocities of the slow waves
through the nine samples were calculated via Eq. (2) and
tabulated in Table III. The propagating speeds of all samples
are similar with an average of 1469 m=s and are less than the
acoustic wave speed in water (1480 m=s).
To further examine the ultrasonic properties of the foam
samples, we performed spectral analysis on the full wave
pulse but focused on the frequency range where the slow
waves dominated. Figure 5 shows the phase velocity versus
frequency curves for all samples for the 0.7–1.3 MHz frequency range. The selection of the frequency range was
based on the full-width-half-maximum (FWHM) of the sample amplitude spectra. The reproducibility for phase velocities was 0.13% calculated by dividing the standard
deviation with the corresponding mean value (Giraudeau
et al., 2003). All phase velocities are lower than the acoustic
velocity in water, which is consistent with the group velocity
measurements and further confirms the dominance of slow
wave propagation through the samples. For the same PPI
samples, the phase velocity decreases with increasing AVF.
Furthermore, the majority of the samples exhibit positive
dispersion (phase velocity increasing with frequency). Table
III lists the dispersion values for the phase velocity curves.
The 20 PPI-12% AVF sample (#9) has the largest dispersion
(15.01 6 2.81 m=s=MHz). There are two exceptions; the 10
PPI-5% AVF sample (#4) and 20 PPI-5% AVF sample (#7)
display minor negative dispersion (phase velocity decreasing
with frequency). Their dispersion values are 1.52 6 1.62
m=s=MHz and 2.9861.44 m=s=MHz, respectively.
Figure 6 displays the total attenuation versus frequency
curves. The attenuation was not corrected for the transmission loss at the foam-water interfaces. The transmission
losses at the interfaces were determined experimentally; the
values were small, ranging between 0.005 and 0.015 dB=cm
and were ignored. The reproducibility for the attenuation
was 10.3%. Generally speaking, attenuation increases with
PPI; for each PPI, attenuation increases with AVF; for the 5
PPI, the 8% AVF (#2) and 12% AVF (#3) samples have similar attenuation behavior within the frequency range of interest but different nBUA values, which are 5.5 6 1.0
dB=MHz=cm and 6.5 6 0.7 dB=MHz=cm, respectively
(Table III). For the 5 PPI and 10 PPI samples, attenuation
increases quasi-linearly with increasing frequency. However,
the 20 PPI samples display a fairly constant attenuation level
with small variations (less than 1.5 dB=cm).
To investigate the influence of foam structure on the
acoustic properties, the phase velocity and attenuation at 1.0
MHz for all samples are plotted in Figs. 7(a) and 7(b),
respectively. At fixed PPI, phase velocity decreases with
FIG. 6. The total attenuation versus frequency curves: (a) 5 PPI, (b) 10 PPI,
and (c) 20 PPI samples. Attenuation was not corrected for transmission loss,
which was experimentally determined to be insignificant.
and 0.5 dB=cm). Second, for the 8% AVF samples (#2, #5,
and #8), the phase velocities of 5 PPI (#2) and 10 PPI (#5)
samples are the same (less than 1 m=s difference), whereas
20 PPI sample (#8) has a smaller phase velocity. Among the
3322
J. Acoust. Soc. Am., Vol. 129, No. 5, May 2011
three 8% AVF samples, the 5 PPI sample (#2) has the largest
attenuation and the 10 PPI sample (#5) has the smallest
attenuation. Third, for the 12% AVF samples (#3, #6, and
#9), the 20 PPI sample (#9) exhibits a slowest phase velocity
while suffers the largest attenuation (12.5360.89 dB=cm).
The phase velocities of 5 PPI (#3) and 10 PPI (#6) samples
have minor difference. The same is true for their attenuation.
Biot’s theory was used to predict the slow wave velocities in the aluminum foams. The Biot’s equations for the
wave speeds propagating in the fluid-filled aluminum foams
are given in details in Appendix A. Tables I and II provide
the tortuosity and other parameters required for the computation. The root-mean-squared error (RMSE) was used to measure the difference between the predictions and experiments.
The RMSE was obtained by first summing the squared errors
of all nine samples between the measured wave speed and
the predicted value and then taking the square root of the
sum. Figure 8 shows a RMSE curve between the predicted
and experimental velocities for n ranging from 1 to 2. The
four n-values published in the literature: 1 (Gibson and
Ashby, 1999), 1.23 (Williams, 1992), 1.46 (Hosokawa and
Zhang et al.: Dispersion and attenuation in bone phantoms
Author's complimentary copy
FIG. 7. The influence of PPI and AVF on the (a) phase velocity and (b)
attenuation at 1.0 MHz.
Otani, 1998), and 2 (Gibson and Ashby, 1999) are also labeled in the figure.
IV. DISCUSSION
We studied nine aluminum foam samples with different
AVFs and PPIs as candidates to mimic cancellous bone
structures. These samples have an open-celled structure
formed by an interconnected network of aluminum ligaments with a comparatively low AVF ranging from 5% to
12%. The cancellous bone has a plate-like structure at high
densities and an open-celled rod-like structure at low densities (Gibson and Ashby, 1999; Hosokawa, 2005). The networks of the aluminum foams in this study have high
resemblance to the low-density human cancellous bone.
Moreover, the majority of these foam samples have ligament
thickness (Al.Th) and separation (Al.Sp) close to the
reported trabecular thickness (Tb.Th ¼ 0.082 0.284 mm)
and separation (Tb.Sp ¼ 0.454 1.307 mm), respectively
(Hildebrand et al., 1999; Ulrich et al., 1999). This is valid
for the 20 PPI samples (see Table I) and the structural factors
or tortuosities of the 20 PPI samples measured in our previous report (Le et al., 2010) are very similar to the measurements of human vertebral cancellous bone (1.04 6 0.004)
using the same air-borne ultrasound approach (Nicholson
and Strelitzki, 1999). However, aluminum has larger elastic
modulus and density (Table II) than those of trabeculae
(Lang, 1970; Williams, 1992; Keavery et al., 1993; Hosokawa and Otani, 1997), which results in an increased acoustic impedance contrast with saturating fluid and enhances the
scattering processes among the ligament-water interfaces.
Biot (1956a,b) predicted two compressional waves
propagating through fluid-saturated porous materials. They
are the fast and slow compressional waves which are associated to the in-phase and out-of-phase motion between the
solid and fluid components, respectively. These two types of
waves are nondispersive when the frequency satisfies the
condition x >> 2g=qfa2, where g is the dynamic viscosity
J. Acoust. Soc. Am., Vol. 129, No. 5, May 2011
Zhang et al.: Dispersion and attenuation in bone phantoms
3323
Author's complimentary copy
FIG. 8. The RMSE between the experimental and Biot’s velocities of the
slow waves for n varying from 1 to 2. The four n-values published in the literature: 1 (Gibson and Ashby, 1999), 1.23 (Williams, 1992), 1.46 (Hosokawa and Otani, 1998), and 2 (Gibson and Ashby, 1999) are labeled.
of water, qf is the density of water, and a is the pore size
(Plona and Johnson, 1980; Johnson and Plona, 1982).
Reports on the fast and slow waves in porous solids are
abundant but similar reports in cancellous bone are not many
(Lakes et al., 1983; Hosokawa and Otani, 1997, 1998;
Hughes et al., 1999; Nicholson and Strelitzki, 1999; Hoffmeister et al., 2000; Mohamed et al., 2003; Mizuno et al.,
2009). Ji et al. (1998) have successfully observed a fast
wave and a slow wave with comparable amplitudes when a
1.0 MHz ultrasound pulse was transmitted along the pore
elongation of various 40 PPI water-saturated aluminum
foams. The pore orientation of the samples was later provided by the ERG Materials and Aerospace Corporation,
Oakland, CA. In the study of Ji et al. (1998) they found the
experimental velocities and Biot’s predictions had good
agreement for n ¼ 2. In our experiments, a strong slow wave
pulse was observed and the fast wave could only be seen after 100-folded amplification. Given the values of the
dynamic viscosity, g, and density, qf of water listed in Table
II, the viscous skin depth d ¼ (2g=qfx)1=2 at 1.0 MHz is 0.18
lm, much smaller than the smallest pore size (1.11 mm) of
our samples. For these pores, considerable amount of the
fluid moved out-of-phase with the solid and consequently,
the generation of slow wave was greatly encouraged (Johnson et al., 1982). Also at 1.0 MHz, the waves are nondispersive according to the condition stated earlier.
The calculated group velocities of slow waves range
between 1463 6 2.4 m=s and 1473 6 2.8 m=s with an average of 1469 m=s and are consistent with the slow wave phase
velocities at 1.0 MHz, which range between 1461 6 1.2 m=s
and 1473 6 0.6 m=s. The discrepancies between the two velocity measurements are small (less than 0.4% or 6 m=s)
because the dispersion of the samples is very minor. These
values fall below the acoustic wave speed of water as
expected
slow wave characteristics (Vslow ¼ Vwater =
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifor
ffi
tortuosity, Johnson et al., 1982; Plona, 1980) and within
the slow wave velocities (1450–1490 m=s) observed in low
volume fraction, (VF 20%) of cancellous bone (Cardoso
et al., 2003). Seven of the nine samples (comprising 78% of
the samples) exhibit positive dispersion ( 1.12 m=s=MHz)
while the other two samples (#4 and #7) display minor negative dispersion. The abnormal dispersion was first observed
in cancellous bone by Nicholson et al. (1996). The phenomenon illustrates the co-existence of several possible mechanisms in addition to absorption when the ultrasound beams
interact with the foam samples. The possible causes are the
scattering effects (Nicholson et al., 1996) including multiple
scattering (Haı̈at et al., 2008b) and the interference between
the fast and slow waves (Marutyan et al., 2006; Anderson et
al., 2008; Bauer et al., 2008) even though the fast waves in
our case have extremely small amplitudes.
According to the product information (ERG Materials
and Aerospace Corporation, Oakland CA ), the DUOCEL
aluminum foam is formed by an array of bubbles with each
bubble being a 14-faceted polyhedron. Once solidified, the
foam will leave 14 windows in the 14 facets and aluminum
struts as the window frame to create an open cell. Each
strut shares between three adjacent bubbles to create characteristic triangular cross sections. The VF determines the
3324
J. Acoust. Soc. Am., Vol. 129, No. 5, May 2011
larger resistance to compression and deformation. Second,
the methodology of the experiments was different. We used
ultrasound to create mechanical stresses to the structure
while Gibson and Ashby (1999) used mechanical bending.
By means of curve-fitting to the experimental data, Williams
(1992) found n ¼ 1.23 with ultrasound beam parallel to the
columnar orientation of the bovine trabeculae and n ¼ 2.35
for a random structure. Hosokawa and Otani (1998) used bovine cancellous bone (porosity b ¼ 0.82) and found n ¼ 1.46
and n ¼ 2.14 depending on whether the beam was parallel or
perpendicular to the trabecular alignment. The measured
wave speeds were compared with Biot’s predictions using a
range of n-values ranging from 1 to 2 and the RMSE is
shown in Fig. 8. The four n-values: 1, 1.23, 1.46, and 2 are
also labeled in the figure. The n-value of 2 provided the
worst prediction with a RMSE of 172 m=s or 12%. The predictions by the first three n-values conform comparatively
well to the experimental results with RMSE of 8.8 m=s
(n ¼ 1), 10.7 m=s (n ¼ 1.23), and 14.8 m=s (n ¼ 1.46),
respectively. At n ¼ 1.5, the RMSE is 16 m=s or 1%. Below
this value, the RMSE declines mildly to 0.6% at n ¼ 1 and
above it, the RMSE increases sharply and amounts up to
12% at n ¼ 2. Except for one sample (#1), the wave speeds
of all samples were under-predicted by the theory. The main
reasons for the discrepancies could be due to the more complex porous structure of the foams than the Biot’s porous
materials and the presence of the scattering processes which
were not accounted for by the theory.
V. CONCLUSIONS
The ultrasound properties of water-saturated aluminum
foams were studied experimentally. The ligament thickness
and pore size of the foam samples are very similar to those
of human cancellous bones. The ligament network has close
similarity to the low-density cancellous bone with opencelled rod-like structures. Two waves, one fast and one slow,
were observed but the fast wave could not been seen without
significant amplification (100). The strong non-dispersive
slow waves were consistently observed in all samples and
were confirmed as Biot’s slow waves because the propagation speeds were smaller than the acoustic sound speed in
water. The slow wave speeds decreased with AVF and PPI,
respectively. The attenuation increased with AVF and generally with PPI except for the 20 PPI samples where significant
scattering and pulse-induced volume averaging might take
place. The experimental and theoretical speeds were in
agreement with a RMSE of 16 m=s (1%) for n equal to 1.5.
Above this value, the RMSE rose sharply. We conclude that
the aluminum foams especially the 20 PPI samples have similar structural and ultrasonic properties to those of cancellous
bone and thus have a good potential to become a bone-mimicking phantom. Future studies such as phantom anisotropy
can further enhance our understanding of the phantom
properties.
ACKNOWLEDGMENTS
This work was supported by a Discovery Grant (LL)
from the Natural Sciences and Engineering Research Council
Zhang et al.: Dispersion and attenuation in bone phantoms
Author's complimentary copy
cross-sectional shape and actual size of the strut. The two
foam parameters, PPI and AVF, play an important role in the
ultrasonic propagation through the samples. The works by
many investigators such as Hosokawa and Otani (1998) have
shown that the fast and slow waves display opposite behavior with VF. Our samples show the velocities of slow waves
decrease with increasing VF (or decreasing porosity), which
agrees well with the reported slow wave observations with
porosity (Plona, 1980; Cardoso et al., 2003; Mohamed et al.,
2003; Hosokawa, 2008). As PPI increases, both the cell and
pore sizes decrease. The narrowing of the pores might
restrict the fluid motion and increase friction of propagation,
thus increasing attenuation of the slow wave and reducing
the speed. At fixed PPI, as AVF increases, the same narrowing of the pores occurs. The effects of PPI and AVF on
attenuation and wave speed are more prominent for 20 PPI
where the wavelength is comparable to the pore size.
The attenuation generally increases with AVF at fixed
PPI. The attenuation of 5 PPI and 10 PPI samples increases
with frequency, whereas the attenuation of the 20 PPI samples with frequency is less obvious and exhibits a nonincreasing attenuation level within the main frequency band.
The 20 PPI samples have more pores and as a consequence,
a more number of scattering interfaces. This enhances the
scattering processes among ligaments within the samples,
giving rise to the low intensity random signals affecting the
coherent signals and coda. At 20 PPI, the cell size (1.27 mm)
and pore size (1.11–1.20 mm) are smaller and comparable to
the dominant wavelength (1.5 mm). More volume averaging due to a broad pulse is expected when ultrasound beam
passes through the samples and thus generates a smoother
transmitted signal. Both the scattering and volume averaging
are perhaps the dominant processes contributing to the nonincreasing attenuation behavior of the 20 PPI samples. The
average attenuation of the 20 PPI-12% AVF sample at 1.0
MHz was 12.53 dB=cm and was within 10.5–15.0 dB=cm
range as reported in the literature (Strelitzki and Evans,
1996; Hosokawa and Otani, 1998; Chaffai et al., 2000;
Mujagic et al., 2008).
Biot’s theory has been applied with some success to predict propagating velocities in cancellous bone (Haire and
Langton, 1999). According to Gibson and Ashby (1999), the
value of exponent parameter n in Eq. (A9) depends on the
structural type of the solid frame and direction of force
loading. They modeled cancellous bones as low- and highdensity equiaxed structures, prismatic-celled structures, and
plate structures. By means of mechanical testing, they
obtained n ¼ 1 for prismatic-celled structure when the load
was along the prism axes and for plate-like structure when
the stress was parallel to the plates. The exponent could go
up to 2 when the increased perforation in the plate model
turned the structure into a low-density equiaxed structure,
which is open-celled with a network of rods. Higher exponent such as 3 could be obtained when the stresses were either applied across the prism axes or normal to the plates.
Our experiment was different from those of Gibson and
Ashby (1999) in two fundamental ways. First, the tested material was different. The DUOCEL foam structure is 14-facet
polyhedral with triangular cross-section, which might have
APPENDIX A: FAST AND SLOW WAVE VELOCITIES IN
BIOT’S POROUS MATERIALS
Following Williams (1992), the velocities of the fast
and slow compressional waves travelling in liquid-filled porous materials are given, respectively, by
2
¼
Vfast;slow
2ðPR Q2 Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
D6 D2 4ðPR Q2 Þ q11 q22 q212
(A1)
and
D ¼ Pq22 þ Rq11 2Qq12
(A2)
where the parameters P, Q, and R are the modified elastic moduli:
P¼
b KKfs 1 Kb þ b2 Ks þ ð1 2bÞðKs Kb Þ
1 b KKbs þ b KKfs
Q¼
R¼
1 b KKbs bKs
þ
4l
; (A3)
3
;
(A4)
Ks b2
;
1 b KKbs þ b KKfs
(A5)
1 b KKbs þ b KKfs
b is the porosity, Kf is the bulk modulus of the fluid, Ks is the
bulk modulus of the solid, Kb and l are the bulk and shear
moduli of the skeletal frame. The terms q11, q12, and q22
describe densities arising from the relative motion between
solid and fluid and the interaction due to the inertial coupling:
q11 þ q12 ¼ ð1 bÞqs ;
(A6)
q12 þ q22 ¼ bqf ;
(A7)
q12 ¼ ð1 aÞbqf
(A8)
and
where a is the tortuosity.
For the cellular foam, the elastic moduli of the solid Es
and the frame E* are related by a power law relationship
(Williams, 1992)
E ¼ Es ð1 bÞn
(A9)
where n varies from 1 to 3 depending on the geometric structure of the solid frame and the ligament alignment with
respect to the incident ultrasound beam (Gibson and Ashby,
1999). The moduli Ks, Kb, l, Es, and E* are related via the
J. Acoust. Soc. Am., Vol. 129, No. 5, May 2011
Poisson’s ratio of the solid (ts) and the frame (tb), respectively (Williams, 1992)
Ks ¼ Es =3ð1 2ts Þ;
(A10)
Kb ¼ E =3ð1 2tb Þ;
(A11)
l ¼ E=2ð1 þ tb Þ
(A12)
and
where ts ¼ 0.33 (MatWeb) and tb ¼ 0.33 for open-celled
foam (Gibson and Ashby, 1999).
Anderson, C. C., Marutyan, K. R., Holland, M. R., Wear, K. A., and Miller,
J. G. (2008). “Interference between wave modes may contribute to the
apparent negative dispersion observed in cancellous bone,” J. Acoust. Soc.
Am. 124, 1781–1789.
Bauer, A. Q., Marutyan, K. R., Holland, M. R., and Miller, J. G. (2008).
“Negative dispersion in bone: the role of interference in measurements of
the apparent phase velocity of two temporally overlapping signals,” J.
Acoust. Soc. Am. 123, 2407–2414.
Biot, M. A. (1956a). “Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range,” J. Acoust. Soc. Am. 28, 168–
178.
Biot, M. A. (1956b). “Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range,” J. Acoust. Soc. Am. 28,
179–191.
Bossy, E., Laugier, P., Peyrin, F., and Padilla, F. (2007). “Attenuation in trabecular bone: A comparison between numerical simulation and experimental results in human femur,” J. Acoust. Soc. Am. 122, 2469–2475.
Cardoso, L., Teboul, F., Sedel, L., Oddou, C., and Meunier, A. (2003). “In
vitro acoustic waves propagation in human and bovine cancellous bone,”
J. Bone Miner. Res. 18, 1803–1812.
Chaffai, S., Padilla, F., Berger, G., and Laugier, P. (2000). “In vitro measurement of the frequency-dependent attenuation in cancellous bone
between 0.2 and 2 MHz,” J. Acoust. Soc. Am. 108, 1281–1289.
Clarke, A. J., Evans, J. A., Truscott, J. G., Milner, R., and Smith, M. A.
(1994). “A phantom for quantitative ultrasound of trabecular bone,” Phys.
Med. Biol. 39, 1677–1687.
Cobbold, R. S. C. (2008). Foundations of Biomedical Ultrasound (Oxford
University Press, New York), Chap. 2, pp. 96–134.
Droin, P., Berger, G., and Laugier, P. (1998). “Velocity dispersion of acoustic waves in cancellous bone,” IEEE Trans. Ultrason. Ferroelectr. Freq.
Control 45(3), 581–592.
ERG Materials and Aerospace Corporation, Oakland, CA. Available
at < http://www.ergaerospace.com/foamproperties/introduction.htm> (Last
viewed August 14, 2010).
Gibson, J. L. (1985). “The mechanical behavior of cancellous bone,” J. Biomech. 18, 317–328.
Gibson, J. L., and Ashby, F. M. (1999). Cellular Solids (Cambridge University Press, London, UK), pp. 429–452.
Giraudeau, B., Gomez, M. A., and Defontaine, M. (2003). “Assessing the
reproducibility of quantitative ultrasound parameters with standardized
coefficient of variation or intraclass correlation coefficient: A unique
approach,” Osteoporosis Int. 14(7), 614–615.
Haı̈at, G., Padilla, F., Peyrin, F., and Laugier, P. (2008a). “Fast wave ultrasonic propagation in trabecular bone: Numerical study of the influence of
porosity and structural anisotropy,” J. Acoust. Soc. Am. 123, 1694–1705.
Haı̈at, G., Lhemery, A., Renaud, F., Padilla, F., Laugier, P., and Naili, S.
(2008b) “Velocity dispersion in trabecular bone: Influence of multiple
scattering and of absorption,” J. Acoust. Soc. Am. 124(6), 4047–4058.
Haire, T. J., and Langton, C. M. (1999). “Biot theory: A review of its application to ultrasound propagation through cancellous bone,” Bone 24, 291–
295.
Hildebrand, T., Laib, A., Muller, R., Dequeker, J., and Ruegsegger, P.
(1999). “Direct three-dimensional morphometric analysis of human cancellous bone: Microstructural data from spine, femur, iliac crest, and
calcaneus,” J. Bone Miner. Res. 14, 1167–1174.
Hoffmeister, B. K., Whitten, S. A., and Rho, J. Y. (2000). “Low-megahertz
ultrasonic properties of bovine cancellous bone,” Bone 26, 635–642.
Zhang et al.: Dispersion and attenuation in bone phantoms
3325
Author's complimentary copy
of Canada and Alberta Health Services. We want to thank S.
Dyer and G. Bornkamp of ERG Materials and Aerospace
Corporation, Oakland, CA for providing the pore orientation
and information of the foams used in our experiments.
3326
J. Acoust. Soc. Am., Vol. 129, No. 5, May 2011
Mujagic, M., Ginsberg, H. J., and Cobbold, R. S. C. (2008). “Development
of a method for ultrasound-guided placement of pedicle screws,” IEEE
Trans. Ultrason. Ferroelectr. Freq. Control 55, 1267–1276.
Nguyen, T. V., Center, J. R., and Eisman, J. A. (2004). “Osteoporosis:
Underrated, underdiagnosed and undertreated,” Med. J. Aust. 180, S18–
S22.
Nicholson, P. H., Lowet, G., Langton, C. M., Dequeker, J., and Van der
Perre, G. (1996). “A comparison of time-domain and frequency-domain
approaches to ultrasonic velocity measurement in trabecular bone,” Phys.
Med. Biol. 41, 2421–2435.
Nicholson, P. H., and Strelitzki, R. (1999). “Ultrasonic slow waves in airsaturated cancellous bone,” Ultrasonics 37, 445–449.
Plona, T. J. (1980). “Observation of a second bulk compressional wave in a
porous medium at ultrasonic frequencies,” Appl. Phys. Lett. 36, 259–261.
Plona, T. J., and. Johnson, D. (1980). “Experimental study of the two bulk
compressional modes in water-saturated porous structures,” Ultrason.
Symp. Proc. 1, 868–872.
Rossman, P., Zagzebski, J., Mesina, C., Sorenson, J., and Mazess, R. (1989).
“Comparison of speed of sound and ultrasound attenuation in the os calcis
to bone density of radius, femur and lumbar spine,” Clin. Phys. Physiol.
Meas. 10, 353–360.
Sachse, W., and Pao, Y. H. (1977). “On the determination of phase and
group velocities of dispersive waves in solids,” J. Appl. Phys. 49, 4320–
4327.
Silverman, S. L. (2006). “Lessons from osteoporosis clinical trials,” Alzheimer’s Dementia 2, 155–159.
Strelitzki, R., and Evans, J. A. (1996). “On the measurement of the velocity
of ultrasound in the os calcis using short pulses,” Eur. J. Ultrasound 4,
205–213.
Strelitzki, R., Evans, J. A., and Clarke, A. J. (1997). “The influence of porosity and pore size on the ultrasonic properties of bone investigated using a
phantom material,” Osteoporosis Int. 7, 370–375.
Ulrich, D., Rietbergen, B. V., Laib, A., and Ruegsegger, P. (1999). “The
ability of three-dimensional structural indices to reflect mechanical aspects
of trabecular bone,” Bone 25, 55–60.
Verhoef, W. A., Cloostermans, M. J. T. M., and Thijssen, J. M. (1985).
“Diffraction and dispersion effects on the estimation of ultrasound attenuation and velocity in biological tissues,” IEEE Trans. Biomed. Eng. 32,
521–529.
Wear, K. A. (2005). “The dependencies of phase velocity and dispersion on
trabecular thickness and spacing in trabecular bone-mimicking phantoms,”
J. Acoust. Soc. Am. 118, 1186–1192.
Wear, K. A. (2008a). “Ultrasonic attenuation in parallel-nylon-wire cancellous-bone-mimicking phantoms,” J. Acoust. Soc. Am. 124, 4042–4046.
Wear, K. A., and Harris, G. R. (2008b). “Frequency dependence of backscatter from thin, oblique, finite-length cylinders measured with a focused
transducer-with applications in cancellous bone,” J. Acoust. Soc. Am.
124, 3309–3314.
Wear, K. A. (2009). “The dependencies of phase velocity and dispersion on
volume fraction in cancellous bone-mimicking-phantoms,” J. Acoust. Soc.
Am. 125, 1197–1201.
Weaver, R. (1998). “Ultrasonics in an aluminum foam,” Ultrasonics 36,
435–442.
Werner, P. (2005). “Knowledge about osteoporosis: Assessment, correlates
and outcomes,” Osteoporosis Int. 16, 115–127.
Westmacott, C. F. (1995). “Osteoporosis-what is it, how is it measured and
what can be done about it?,” Radiography 1, 35–47.
Williams, J. L. (1992). “Ultrasonic wave propagation in cancellous and cortical bone: prediction of some experimental results by Biot’s theory,” J.
Acoust. Soc. Am. 91(2), 1106–1112.
Zhang et al.: Dispersion and attenuation in bone phantoms
Author's complimentary copy
Hosokawa, A., and Otani, T. (1997). “Ultrasonic wave propagation in bovine cancellous bone,” J. Acoust. Soc. Am. 101, 558–562.
Hosokawa, A., and Otani, T. (1998). “Acoustic anisotropy in bovine cancellous bone,” J. Acoust. Soc. Am. 103, 2718–2722.
Hosokawa, A. (2005). “Simulation of ultrasound propagation through bovine cancellous bone using elastic and Biot’s finite-difference time-domain methods,” J. Acoust. Soc. Am. 118(3), 1782–1789.
Hosokawa, A. (2008). “Development of a numerical cancellous bone model
for finite-difference time-domain simulations of ultrasound propagation,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control 55, 1219–1233.
Hughes, E. R., Leighton, T. G., Petley, G. W., and White, P. R. (1999).
“Ultrasonic propagation in cancellous bone: A new stratified model,”
Ultrason. Med. Biol. 25, 811–821.
Ji, Q., Le, L. H., Filipow, L. J., and Jackson S. A. (1998). “Ultrasonic wave
propagation in water-saturated aluminum foams,” Ultrasonics 36, 759–765.
Johnson, D. L., Plona, T. J., and Scala, C. (1982). “Tortuosity and acoustic
slow waves,” Phys. Rev. Lett. 49(25), 1840–1844.
Johnson, D. L., and Plona, T. J. (1982). “Acoustic slow waves and the consolidation transition,” J. Acoust. Soc. Am. 72, 556–565.
Keaveny, T. M., and Hayes, W. C. (1993). “A 20-year perspective on the
mechanical properties of trabecular bone,” J. Biomech. Eng. 115, 534–542
Lakes, R., Yoon, H. S., and Katz, J. L. (1983). “Slow compressional wave
propagation in wet human and bovine cortical bone,” Science 220, 513–515.
Lang, S. B. (1970). “Ultrasonic method for measuring elastic coefficients of
bone and results on fresh and dried bovine bones,” IEEE Trans. Biomed
Eng. 17, 101–105.
Langton, C. M., Palmer, S. B., and Porter, R. W. (1984). “The measurement
of broadband ultrasonic attenuation in cancellous bone,” Eng. Med. 13,
89–91.
Langton, C. M., and Njeh, C. F. (2004). The Physical Measurement of Bone
(IOP Publishing, London), Chap. 14, pp. 412–464.
Le, L. H. (1998). “An investigation of pulse-timing techniques for broadband ultrasonic velocity determination in cancellous bone: A simulation
study,” Phys. Med. Biol. 43, 2295–2308.
Le, L. H., Zhang, C., Ta, D., and Lou, E. (2010). “Measurement of tortuosity
in aluminum foams using airborne ultrasound,” Ultrasonics 50, 1–5.
Lee, K. I., and Choi, M. J. (2009). “Phase velocity and normalized broadband ultrasonic attenuation in polyacetal cuboid bone-mimicking
phantoms,” J. Acoust. Soc. Am. 121, EL263–EL269.
Legrand, E., Chappard, D., Pascaretti, C., Duquenne, M., Krebs, S., Rohmer,
V., Basle, M. F., and Audran, M. (2000). “Trabecular bone nicroarchitecture, bone mineral density, and vertebral fractures in male osteoporosis,”
J. Bone Miner. Res.15, 13–19.
Lin, J. T., and Lane, J. M. (2004). “Osteoporosis: A review,” Clin. Orthop.
Relat. Res. 425, 126–134.
Lobkis, O. I., and Weaver, R. L. (2001). “Mode counts in an aluminum
foam,” J. Acoust. Soc. Am. 109, 2636–2641.
Lu, T. J., Hess, A., and Ashby, M. F. (1999). “Sound absorption in metallic
foams,” J. Appl. Phys. 85, 7528–7539.
Marutyan, K. R., Holland, M. R., and Miller, J. G. (2006). “Anomalous negative dispersion in bone can result from the interference of fast and slow
waves,” J. Acoust. Soc. Am. 120, EL55–EL61.
MatWeb, Online materials information resource at http://www.matweb.com
(Last viewed May 1, 2010).
Mizuno, K., Matsukawa, M., Otani, T., Laugier, P., and Padilla, F. (2009).
“Propagation of two longitudinal waves in human cancellous bone: An in
vitro study,” J. Acoust. Soc. Am. 125, 3460–3466.
Mohamed, M. M., Shaat, L. T., and Mahmoud, A. N. (2003). “Propagation
of ultrasonic waves through demineralized cancellous bone,” IEEE Trans.
Ultrason. Ferroelectr. Freq. Control 50, 279–288.